From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1942 Path: news.gmane.org!not-for-mail From: Martin Escardo Newsgroups: gmane.science.mathematics.categories Subject: re: Limits Date: Thu, 3 May 2001 13:59:08 +0100 (BST) Message-ID: <15089.14134.867261.350412@henry.cs.bham.ac.uk> References: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018221 1310 80.91.229.2 (29 Apr 2009 15:17:01 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:17:01 +0000 (UTC) To: Category Mailing List Original-X-From: rrosebru@mta.ca Thu May 3 15:31:32 2001 -0300 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f43I4pF03260 for categories-list; Thu, 3 May 2001 15:04:51 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f In-Reply-To: X-Mailer: VM 6.43 under 20.4 "Emerald" XEmacs Lucid Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 10 Original-Lines: 78 Xref: news.gmane.org gmane.science.mathematics.categories:1942 Archived-At: Tobias Schroeder writes: > - Can the limit of a sequence of real numbers be expressed > as a categorical limit (of course it can if the sequence is > monotone, but what if it is not)? I think I have an answer to this question (without cheating). It may be well known or wrong (I haven't carefully checked the details, but I believe that they are correct). Given a metric space X with distance function d, construct a category, also called X, as follows. The objects of the category X are the points of the space X. An element of the hom-set X(x,y) is a triple (r,x,y) with r a real number such that d(x,y)<=r. The composite of the arrows r:x->y and s:y->z is the arrow s+r:x->z. This is well defined by virtue of the triangle inequality d(x,z)<=d(x,y)+d(y,z). By virtue of the condition d(x,x)=0, we have identities. Notice that all arrows are mono. Of course, because the category X is small and it is not a preordered set, it doesn't have all limits. But some limits do exist. CLAIM: Let x_n be a sequence of points of X, and, for each n, let the arrow r_n:x_{n+1}->x_n be d(x_n,x_{n+1}). If the sum of r_k over k>=0 exists, then this omega^op-diagram has a categorical limit. The source of the limiting cone is the metric limit l of the sequence. The projection p_n:l->x_n is the sum of r_k over k>=n. If q_n:m->x_n is another cone, then the mediating map u:l->m exists (and will be automatically unique), because, by definition of cone and of our category, q_n will have to be bigger than r_n, and then u=q_n-r_n does the job. Remarks. (1) For any given Cauchy sequence, one can construct an equivalent Cauchy sequence for which the assumption in the second sentence of the claim fails. Using classical logic, for any given Cauchy sequence, one can construct an equivalent Cauchy sequence for which the assumption holds. (2) In (some flavours of) constructive mathematics, the notion of a Cauchy sequence "with fixed rate of convergence" is taken as basic. This often is taken to mean that d(x_n,x_n+1)<=c^n for a fixed c with 0Y is called non-expansive if d(fx,fx')<=d(x,x'). If the natural numbers are metrized by d(m,n)=c^min(m,n) for m/=n, to get a space N, then such a Cauchy sequence is just a non-expansive map N->X. It converges if and only if the non-expansive map has a non-expansive extension to N_{infty}, the metric completion of N (which, topologically, is the one-point compactification of N). And non-expansive maps are functors---see (3) below. (3) Recall that a map f:X->Y is called lipschitz if there is a constant c for which d(fx,fx')<=c.d(x,x'). A lipschitz map f:X->Y gives rise to a functor f:X->Y defined by f(r:x->x')=c.r:f(r)->f(x'). That is, the object part is given by the map itself, and the arrow part is given by multiplication with the lipschitz coefficient. (4) We have taken the arrows r_n to be d(x_n,x_{n+1}). But actually any choice of arrows does the job, provided the sum of r_k over k>=0 is finite. (5) Other two conditions for the distance function of a metric space, which were not used in the definition of the category X, are (i) d(x,y)=0 implies x=y, and (ii) d(x,y)=d(y,x). By the first, our category is skeletal. By the second, it is selfdual. Of course, people have considered generalized metric spaces in which these are not assumed to hold. See, for example, Lawvere's paper "Metric spaces, generalized logic, and closed categories", in which he regards a generalized metric space X as an enriched category with X(x,y)=d(x,y) (so he has hom-numbers instead of hom-sets). Here we have hom-sets (of numbers). MHE